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Theorem ineccnvmo 36085
 Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.)
Assertion
Ref Expression
ineccnvmo (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem ineccnvmo
StepHypRef Expression
1 relcnv 5944 . . 3 Rel 𝐹
2 id 22 . . . 4 (𝑦 = 𝑧𝑦 = 𝑧)
32inecmo 36083 . . 3 (Rel 𝐹 → (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥))
41, 3ax-mp 5 . 2 (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥)
5 brcnvg 5725 . . . . 5 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝐹𝑥𝑥𝐹𝑦))
65el2v 3417 . . . 4 (𝑦𝐹𝑥𝑥𝐹𝑦)
76rmobii 3314 . . 3 (∃*𝑦𝐵 𝑦𝐹𝑥 ↔ ∃*𝑦𝐵 𝑥𝐹𝑦)
87albii 1821 . 2 (∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥 ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
94, 8bitri 278 1 (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∨ wo 844  ∀wal 1536   = wceq 1538  ∀wral 3070  ∃*wrmo 3073  Vcvv 3409   ∩ cin 3859  ∅c0 4227   class class class wbr 5036  ◡ccnv 5527  Rel wrel 5533  [cec 8303 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-xp 5534  df-rel 5535  df-cnv 5536  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ec 8307 This theorem is referenced by:  ineccnvmo2  36088
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